Let $X \subset \mathbb{A}^n$ be an affine variety. Let us define a $\mathbb{Q}$-open set of X to be a union of sets of the form
$ D(f) = \{x \in X| f(x) \neq 0\},\,\, f \in \mathbb{Q}[x_1,\dots,x_n] $
Let us define the $\mathbb{Q}$-regular functions on a $\mathbb{Q}$-open set $U$ to be functions such that at every $x \in U$ there exists $U_x \subset U$ s.t $x \in U_x$ and $f = \frac{p}{q}$ on $U_x$ where $p,q \in \mathbb{Q}[x_1,\dots,x_n]$. We will denote the $\mathbb{Q}$-regular functions on $U$ by $\mathcal{O}(U)(\mathbb{Q})$
Now we know from standard results that $\mathcal{O}(D(f)) = \mathbb{C}[X]_f = \mathbb{C}\otimes_{\mathbb{Q}}\mathbb{Q}[X]_f$.
We also know that $\mathbb{Q}[X]_f = \mathcal{O}(D(f))(\mathbb{Q})$.
My question is do we know that for an arbitrary $\mathbb{Q}$-open set $U$ that $\mathbb{C}\otimes_{\mathbb{Q}}\mathcal{O}(U)(\mathbb{Q}) = \mathcal{O}(U)$?
Thanks!